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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 9 "Stability" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 390 "This laboratory is \+
almost identical to the lab for the previous module.  Although this mo
dule is essentially theoretical we are interested in theory because of
 what it tells us about particular models.  Thus, you may want to veri
fy the predictions of the theorems in this module experimentally.  Log
istic models are good models to use because it is easy to compute the \+
relevant derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 141 "At the end of this lab there is a cell that illustrat
es the way in which Maple can work with series like the geometric seri
es in this module." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "The cell below defines a procedure  -- " }{TEXT 256 6 "co
bweb" }{TEXT -1 89 "  -- that can be used to draw cobweb diagrams.\nTh
is procedure requires three arguments.\n\n" }{TEXT 257 4 "fcn " }
{TEXT -1 100 " -- the function that describes how each term of the seq
uence is computed from the preceding term.\n\n" }{TEXT 258 5 "first" }
{TEXT -1 50 " -- the value of the first term of the sequence.\n\n" }
{TEXT 259 1 "n" }{TEXT -1 76 " -- the number of terms of the sequence \+
to be shown in the cobweb diagram \n\n" }{TEXT 260 27 "Evaluate the ce
ll below now" }{TEXT -1 4 ".  \n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1488 "with(plots):\nwith(plottools):\n\nPopModel :=\n    \+
 proc(fcn, first, n) \n        if n <= 1 then first else fcn(PopModel(
fcn, first, n-1)) fi:\n     end:\n\ncobweb := \n     proc(fcn, first, \+
n, top)\n     local blk, rd, blu, frm, cobweblist, j;\n\n     cobwebli
st := \{line([first, 0], [first, fcn(first)])\}:\n\n     for j from 3 \+
to n do\n        cobweblist := cobweblist union\n            \{line([P
opModel(fcn, first, j-2), PopModel(fcn, first, j-1)],\n               \+
   [PopModel(fcn, first, j-1), PopModel(fcn, first, j-1)]), \n        \+
     line([PopModel(fcn, first, j-1), PopModel(fcn, first, j-1)],\n   \+
               [PopModel(fcn, first, j-1), PopModel(fcn, first, j)])\}
\n     od:\n\n     frm := display(\{line([0, 0], [0, top]),\n         \+
            line([0, top], [top, top]),\n                     line([to
p, top], [top, 0]),\n                     line([top, 0], [0, 0])\},\n \+
                    color = BLACK,\n                     axes = NONE,
\n                     scaling = CONSTRAINED):\n\n     blk := plot(f(p
),\n                    p = 0..top,\n                    y = 0..top,\n
                    color = BLACK,\n                    scaling = CONS
TRAINED):\n\n     rd  := display(cobweblist,\n                    colo
r = RED,\n                    axes = NONE,\n                    scalin
g = CONSTRAINED):\n\n     blu := display(\{line([0, 0], [top, top])\},
\n                    color = BLUE,\n                    axes = NONE,
\n                    scaling = CONSTRAINED):\n\n     display([blk, rd
, blu, frm]);   \n  end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 133 "The following cell illustrates how this \+
new procedure can used along with other Maple procedures to investigat
e long term behavior.  " }{TEXT 261 16 "Evaluate it now." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 359 " f \+
:= p -> a * (1 - p / c) * p:\n\n a := 3.0:\n c := 1000.0:\n\n  cobweb(
f, 50, 30, 1000);\n\n  pop := proc(n) option remember;\n         if n \+
= 1 then 50 else f(pop(n-1)) fi\n         end:\n\n  for k from 1 to 30
 \n      do printf(`%4d %10.4f`, k, pop(k));\n      print();\n      od
;\n\n  plot([seq([n, pop(n)], n = 1 .. 30)], \n        year = 1 .. 30,
 title = `Population`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 53 "The cell below illustrates the Maple synt
ax for sums." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "evalf(Sum(.5^k, k=0..10));" }}}}{MARK "0 0 0" 9 
}{VIEWOPTS 1 1 0 1 1 1803 }